結果

問題 No.2262 Fractions
ユーザー 👑 Nachia
提出日時 2023-04-08 11:18:11
言語 C++17
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 77 ms / 2,000 ms
コード長 10,712 bytes
コンパイル時間 1,242 ms
コンパイル使用メモリ 94,800 KB
最終ジャッジ日時 2025-02-12 03:49:11
ジャッジサーバーID
(参考情報)
judge2 / judge2
このコードへのチャレンジ
(要ログイン)
ファイルパターン 結果
sample AC * 1
other AC * 45
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ソースコード

diff #
プレゼンテーションモードにする

#line 2 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\divisor-convolution.hpp"
#line 2 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\prime-sieve-explicit.hpp"
#include <vector>
#include <algorithm>
#include <cassert>
#include <iostream>
namespace nachia{
namespace prime_sieve_explicit_internal{
std::vector<bool> isprime = { false }; // a[x] := isprime(2x+1)
void CalcIsPrime(int z){
if((int)isprime.size() *2+1 < z+1){
int new_z = isprime.size();
while(new_z*2+1 < z+1) new_z *= 2;
z = new_z-1;
isprime.resize(z+1, true);
for(int i=1; i*(i+1)*2<=z; i++) if(isprime[i]){
for(int j=i*(i+1)*2; j<=z; j+=i*2+1) isprime[j] = false;
}
}
}
std::vector<int> prime_list = {2};
int prime_list_max = 0;
void CalcPrimeList(int z){
while((int)prime_list.size() < z){
if((int)isprime.size() <= prime_list_max + 1) CalcIsPrime(prime_list_max + 1);
for(int p=prime_list_max+1; p<(int)isprime.size(); p++){
if(isprime[p]) prime_list.push_back(p*2+1);
}
prime_list_max = isprime.size() - 1;
}
}
void CalcPrimeListUntil(int z){
if(prime_list_max < z){
CalcIsPrime(z);
for(int p=prime_list_max+1; p<(int)isprime.size(); p++){
if(isprime[p]) prime_list.push_back(p*2+1);
}
prime_list_max = isprime.size() - 1;
}
}
}
bool IsprimeExplicit(int n){
using namespace prime_sieve_explicit_internal;
if(n == 2) return true;
if(n % 2 == 0) return false;
CalcIsPrime(n);
return isprime[(n-1)/2];
}
int NthPrimeExplicit(int n){
using namespace prime_sieve_explicit_internal;
CalcPrimeList(n);
return prime_list[n];
}
int PrimeCountingExplicit(int n){
using namespace prime_sieve_explicit_internal;
if(n < 2) return 0;
CalcPrimeListUntil(n);
auto res = std::upper_bound(prime_list.begin(), prime_list.end(), n) - prime_list.begin();
return (int)res;
}
// [l, r)
std::vector<bool> SegmentedSieveExplicit(long long l, long long r){
assert(0 <= l); assert(l <= r);
long long d = r - l;
if(d == 0) return {};
std::vector<bool> res(d, true);
for(long long p=2; p*p<=r; p++) if(IsprimeExplicit(p)){
long long il = (l+p-1)/p, ir = (r+p-1)/p;
if(il <= p) il = p;
for(long long i=il; i<ir; i++) res[i*p-l] = false;
}
if(l < 2) for(long long p=l; p<2 && p<r; p++) res[l-p] = false;
return res;
}
} // namespace nachia
#line 4 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\divisor-convolution.hpp"
#line 8 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\divisor-convolution.hpp"
namespace nachia{
template<class Elem>
void DivisorZeta(std::vector<Elem>& a){
using namespace prime_sieve_explicit_internal;
int n = a.size() - 1;
for(int d=2; d<=n; d++) if(IsprimeExplicit(d)) for(int i=1; i*d<=n; i++) a[i*d] += a[i];
}
template<class Elem>
void DivisorInvZeta(std::vector<Elem>& a){
using namespace prime_sieve_explicit_internal;
int n = a.size() - 1;
for(int d=2; d<=n; d++) if(IsprimeExplicit(d)) for(int i=n/d; i>=1; i--) a[i] += a[i*d];
}
template<class Elem>
void DivisorMobius(std::vector<Elem>& a){
using namespace prime_sieve_explicit_internal;
int n = a.size() - 1;
for(int d=2; d<=n; d++) if(IsprimeExplicit(d)) for(int i=n/d; i>=1; i--) a[i*d] -= a[i];
}
template<class Elem>
void DivisorInvMobius(std::vector<Elem>& a){
using namespace prime_sieve_explicit_internal;
int n = a.size() - 1;
for(int d=2; d<=n; d++) if(IsprimeExplicit(d)) for(int i=1; i*d<=n; i++) a[i] -= a[i*d];
}
template<class Elem>
std::vector<Elem> GcdConvolution(std::vector<Elem> a, std::vector<Elem> b){
assert(a.size() == b.size());
assert(1 <= a.size());
DivisorInvZeta(a);
DivisorInvZeta(b);
for(int i=1; i<(int)a.size(); i++) a[i] *= b[i];
DivisorInvMobius(a);
return a;
}
template<class Elem>
std::vector<Elem> LcmConvolution(std::vector<Elem> a, std::vector<Elem> b){
assert(a.size() == b.size());
assert(1 <= a.size());
DivisorZeta(a);
DivisorZeta(b);
for(int i=1; i<(int)a.size(); i++) a[i] *= b[i];
DivisorMobius(a);
return a;
}
}
#line 2 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\floor-sum.hpp"
#line 4 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\floor-sum.hpp"
#include <utility>
namespace nachia{
// a : any value
// mod != 0
std::pair<long long, unsigned long long> SafeDiv(long long a, unsigned long long mod){
using u64 = unsigned long long;
if(a >= 0) return std::make_pair(0, (u64)a);
if(mod >= (u64)1 << 62) return std::make_pair(-1, (u64)a + mod);
long long q = a / mod;
long long m = a % (long long)mod;
if(m){ q--; m += mod; }
return std::make_pair(q, m);
}
unsigned long long nC2Uint64(unsigned long long n){
return (n%2) ? ((n-1)/2*n) : (n/2*(n-1));
}
// n : any
// 1 <= m
// a : any
// b : any
// n * a%m + b%m < 2**64
unsigned long long FloorSumU64Unsigned(
unsigned long long n,
unsigned long long m,
unsigned long long a,
unsigned long long b
){
using u64 = unsigned long long;
assert(1 <= m);
u64 ans = 0;
while(n){
if(a >= m){
ans += a / m * nC2Uint64(n);
a %= m;
}
if(b >= m){
ans += b / m * n;
b %= m;
}
u64 y_max = a * n + b;
if (y_max < m) return ans;
n = y_max / m;
b = y_max % m;
y_max = a; a = m; m = y_max;
}
return ans;
}
// n : any
// 1 <= m
// a : any
// b : any
// (n+1) * m < 2**64
unsigned long long FloorSumU64Signed(
unsigned long long n,
unsigned long long m,
long long a,
long long b
){
using u64 = unsigned long long;
auto ua = SafeDiv(a, m);
auto ub = SafeDiv(b, m);
u64 ans = FloorSumU64Unsigned(n, m, ua.second, ub.second);
ans += ua.first / m * nC2Uint64(n);
ans += ub.first / m * n;
return ans;
}
} // namespace nachia
#line 4 "D:\\Programming\\VSCode\\competitive-cpp\\nachia\\math\\rational-number-search.hpp"
namespace nachia{
class RationalNumberSearch{
public:
RationalNumberSearch(unsigned long long maxVal){
assert(maxVal < 1ull << 63);
mx = maxVal;
}
bool hasNext(){ return state >= 0; }
std::pair<unsigned long long, unsigned long long> getNext() const {
switch(state){
case 0: return { a0+a1, b0+b1 };
case 1: return { a0+tr*a1, b0+tr*b1 };
case 2: return { a1+tr*a0, b1+tr*b0 };
case 3: return { a0+(tl+tr)/2*a1, b0+(tl+tr)/2*b1 };
case 4: return { a1+(tl+tr)/2*a0, b1+(tl+tr)/2*b0 };
}
return {0,0};
}
void give(bool toRight){
int x = toRight ? 1 : 0;
switch(state){
case 0:
tl = 1; tr = 2;
if(a0 + a1 > mx || b0 + b1 > mx){ state = -1; }
else{ state = (toRight ? 1 : 2); }
break;
case 1: case 2:
if(x ^ (2-state)){ state += 2; }
else{ tr *= 2; tl *= 2; }
break;
case 3: case 4:
((x ^ (4-state)) ? tr : tl) = (tl+tr)/2;
break;
}
while(givecheck());
}
private:
using UInt = unsigned long long;
UInt a0=0, b0=1, a1=1, b1=0, tl=0, tr=0, mx;
int state = 0;
bool givecheck(){
auto st = [this](int x){ state = x; return true; };
auto trq = [this](UInt x0, UInt x1) -> bool {
bool f = x0+tr*x1 > mx;
if(f) tr = (mx-x0)/x1 + 1;
return f;
};
bool f = false;
switch(state){
case -1 : break;
case 0:
if(a0 + a1 > mx || b0 + b1 > mx){ state = -1; }
break;
case 1:
if(trq(a0,a1)) f = true;
if(trq(b0,b1)) f = true;
if(f) return st(3);
break;
case 2:
if(trq(a1,a0)) f = true;
if(trq(b1,b0)) f = true;
if(f) return st(4);
break;
case 3:
if(tl + 1 == tr){
a0 += a1 * tl;
b0 += b1 * tl;
return st(0);
}
break;
case 4:
if(tl + 1 == tr){
a1 += a0 * tl;
b1 += b0 * tl;
return st(0);
}
break;
}
return false;
}
};
} // namespace nachia
#line 5 "..\\Main.cpp"
#include <string>
#line 8 "..\\Main.cpp"
#include <atcoder/modint>
using namespace std;
using i32 = int;
using u32 = unsigned int;
using i64 = long long;
using u64 = unsigned long long;
#define rep(i,n) for(int i=0; i<(int)(n); i++)
const i64 INF = 1001001001001001001;
using Modint = atcoder::static_modint<998244353>;
i64 countFracs(i64 N){
vector<i64> cnt(N+1);
for(i64 c=1; c<=N; c++) cnt[c] += c;
nachia::DivisorMobius(cnt);
i64 sumcnt = 0;
for(int i=1; i<=N; i++) sumcnt += cnt[i];
return sumcnt;
}
pair<i64,i64> testcase(){
i64 N, K; cin >> N >> K;
i64 cnt = countFracs(N);
if(cnt * 2 - 1 < K) return {-1,-1};
if(cnt == K) return {1,1};
bool sw = false;
if(cnt < K){ K = cnt*2 - K; sw = true; }
vector<i64> quotients = {0};
for(i64 k=1; k*k<N; k++) quotients.push_back(k);
int qp1 = quotients.size();
for(i64 k=1; k*k<=N; k++) quotients.push_back(N/k);
reverse(quotients.begin() + qp1, quotients.end());
int numQ = quotients.size()-1;
vector<i64> mobius(N+1); mobius[1] = 1;
nachia::DivisorMobius(mobius);
vector<i64> mertens = mobius;
rep(i,N) mertens[i+1] += mertens[i];
auto srch = nachia::RationalNumberSearch(N);
i64 a = 0, b = 0;
i64 t = 0;
while(srch.hasNext()){
t++;
i64 ax = srch.getNext().first;
i64 bx = srch.getNext().second;
i64 sumcnt = 0;
rep(q,numQ){
i64 times = mertens[quotients[q+1]] - mertens[quotients[q]];
cnt = nachia::FloorSumU64Signed(quotients[numQ-q]+1, bx, ax, 0);
sumcnt += times * cnt;
}
if(K <= sumcnt){ a = ax; b = bx; }
srch.give(sumcnt < K);
}
if(sw) swap(a, b);
return {a,b};
}
int main(){
int T; cin >> T;
rep(t,T){
auto ans = testcase();
if(ans.first < 0) cout << "-1\n"; else cout << ans.first << '/' << ans.second << '\n';
}
return 0;
}
struct ios_do_not_sync{
ios_do_not_sync(){
ios::sync_with_stdio(false);
cin.tie(nullptr);
}
} ios_do_not_sync_instance;
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